How to improve a trained model over time (i.e. with more predictions)?
That is exactly a neural network works like.
Suppose you have a 1000 examples. How you train a network is: First, you divide these 1000 into maybe 100 batches (10 each). After that's done, you feed a batch to the network get its output and compare it with the ground truth, whatever is the error gets backpropagated. Then, for the next batch and then another. Once all these batches are done, you say an epoch is over. So, the number of epoch is effectively the number of times the network has seen the whole data.
This is how a neural network gets better.